Many applied problems in the natural sciences can be modeled by partial differential equations (PDEs) with heterogeneous coefficients that rapidly vary at small scales. Numerical homogenization methods based on effective models are an efficient

alternative to standard numerical methods (like the finite element method (FEM)), as they require scale resolution only in a small portion of the computational domain.

In this thesis, we introduced numerical homogenization methods for two different classes of multiscale PDEs. First, we considered linear parabolic advection-diffusion problems with highly oscillating data, large Péclet number and compressible velocity fields, e.g., transport processes in porous media. The numerical upscaling strategy appropriately models the effects of the highly oscillating velocity field on the effective diffusion (enhanced or depleted diffusion). As the method is based on a discontinuous Galerkin method it further has favorable stability properties.

Second, we proposed new methods for parabolic nonlinear monotone multiscale problems, e.g., ferromagnetism in composite materials. We combined the backward Euler method in time with a finite element heterogeneous multiscale method (FEHMM) in space. The upscaling procedure however consists of nonlinear problems, which can be computationally expensive. As a remedy, we proposed a linearized method, which is much more efficient.